This paper deals with the problem of testing for the presence of autocorrelation in a system of general linear models (Seemingly Unrelated Regressions, SUR) when the model is formulated as a vector autoregression (VAR) with exogenous variables. The solution presented in this paper is a generalization of the h-statistic for the single equation single parameter case given in Durbin (1970)

Dhrymes, Phoebus J.

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 Columbia University  Department of Economics Discussion Paper Series           Testing for Autocorrelation in Systems of Equations     Phoebus J. Dhrymes       Discussion Paper #:0102-67        Department of Economics Columbia University New York, NY 10027 July 2002  Testing for Autocorrelation in Systems ofEquations∗PHOEBUS J. DHRYMESColumbia UniversityJune 2002AbstractThis paper deals with the problem of testing for the presence of au-tocorrelation in a system of general linear models (Seemingly UnrelatedRegressions, SUR) when the model is formulated as a vector autoregres-sion (VAR) with exogenous variables. The solution presented in thispaper is a generalization of the h-statistic for the single equation singleparameter case given in Durbin (1970).KEY WORDS: Autocorrelation, h-test, H2 -test, VAR1 IntroductionConsider the model∗ c©Phoebus J. Dhrymes, 2002Preliminary material; not to be quoted or disseminated without permission of the author.1yt· = yt−1·A+ xt·B + ut·, t = 1, 2, 3, . . . , T, (1)where yt· is an m -element row vector of dependent, and xt· is a k -element vec-tor of independent variables, respectively; ut·, t = 1, 2, . . . , T is the structuralerror vector. We assumei. {ut· : t = 1, 2, 3, . . . , T} is a sequence of independent identically distrib-uted (i.i.d.) random vectors withE ut· = 0, Cov(ut·) = Σ > 0, (2)defined on some probability space ( Ω , A , P ).ii. It is further assumed thatplimT→∞X ′XT=Mxx > 0, (3)and that the elements in X and U are mutually independent.iii. The system of Eq. (1) is stable, i.e. the characteristic roots of A are lessthan one in absolute value.Regarding the errors, the alternative hypothesis we entertain isut· = ut−1·R + t·. (4)We require, for stationarity, the following assumptions:1. The matrix R is non-singular and stable, i.e. its characteristic roots areless than one in absolute value;2. With little loss of generality, and certainly no loss of relevance, we furtherassume that the matrix R is diagonalizable, i.e. it has the representationR = PΛP−1 , where Λ is the (diagonal) matrix of its characteristic roots.This problem, for the case m = 1 , (and R a scalar) was dealt with by Durbin(1970). A search of widely used econometrics textbooks such as Greene (1999)and Davidson and MacKinnon (1993) discloses no mention of its generalizationto VARs.2Remark 1. If one were to write down a VAR one would normally not beconcerned about the behavior of the “error”, since by definition the errors insuch a system are assumed to be i.i.d. If not, one simply specifies a VAR ofa higher order, in empirical applications. Notwithstanding this observation, inmany applied contexts the logic of the economic model requires the presence ofa specific number of lagged endogenous variables. In such a case, the problemwe are examining here may arise.Remark 2. When the structural error, ut· , is a first order autoregression,the OLS estimators for the parameters of the model in Eq. (1) are inconsis-tent because of the presence of lagged endogenous variables, which aretherefore correlated with the structural error.Thus, if we suspect that the form given in Eq. (4) may be appropriate, we maywish to test the hypothesisH0 : R = 0 ,as against the alternativeH1 : R 6= 0 ,when least squares (OLS) is used to estimate the unknown parameters of Eq.(1).2 Derivation of the Test StatisticWriting the sample asY = Y−1A+XB + U = ZC + U, Z = (Y−1, X), C = (A′, B′)′, (5)the OLS estimator of C is given byC˜ = (Z ′Z)−1Z ′Y = C + (Z ′Z)−1Z ′U. (6)Assuming that a central limit theorem (CLT), such as the Lindeberg CLT, seeDhrymes (1989), pp. 271 ff, we may write the limiting distribution of the OLS3estimator as √T (c˜− c) d→ N(0,Σ⊗M−1zz ), (7)where c = vec(C) and Mzz = plimT→∞[Z ′Z/T ] .From this it is easily verified that√T (a˜− a) d→ N(0,Σ⊗ S11), where M−1zz =[S11 S12S21 S22], (8)and S11 is the (principal) submatrix of M−1zz , consisting of its first m rowsand columns.LetU˜ = Y − ZC˜ = U − Y−1(A˜− A)−X(B˜ −B), (9)be the matrix of OLS residuals and consider the estimator of RR˜ = (U˜ ′−1U˜−1)−1U˜ ′−1U˜ (10)Using Eq. (10), and omitting terms that converge to zero in probability, wemay write, see Dhrymes (1989), pp 161 ff.1√T (R˜− R) ∼ (U˜ ′−1U˜−1)−11√T(U ′−1U − U ′−1Y−1(A˜− A)), (11)either because Z ′Z/T converges, or because C˜ is consistent and has a welldefined limiting distribution, or both. Moreover, using the result again, andbearing in mind that1T U′−1Y−1P→ Σwe finally obtain√T (R˜ −R) ∼ Σ−1 1√TU ′−1U −√T (A˜− A). (12)LetM−1zz =[S11 S12S21 S22]=[S1S2],1The notation X ∼W below means X has the same limiting distribution as W .4and note that √T (A˜− A) ∼ S11√TZ ′U.Vectorizing, we have the expression√Tvec(R˜−R) ∼ 1√T((Im ⊗ [Σ−1U ′−1 − S1Z ′])u, u = vec(U), (13)which obeys the conditions of the Lindeberg CLT, see Dhrymes (1989), pp. 271ff. LetAt = σ(us·, s ≤ t), (14)i.e. it is the σ -algebra generated by the u’s up to t . To evaluate the covariancematrix of the limiting distribution of the left member of Eq. (13), we need tofind the expectation of terms like (Im⊗K)uu′(Im⊗K) . We shall do so by firstconditioning with respect to At−1 . Thus we need to evaluateEAt−1E([Im ⊗ (Σ−1U ′−1 − S1Z ′)]uu′[Im ⊗ (U−1Σ−1 − ZS ′1)]|At−1)=EAt−1(Σ⊗ [Σ−1U ′−1U−1Σ−1 − Σ−1U ′−1ZS ′1 − S1Z ′U−1Σ−1 + S1Z ′ZS ′1])=Σ⊗ [Σ−1ΣΣ−1 − Σ−1(Σ, 0)S ′1 − S1(Σ, 0)′Σ−1 + S11]=Σ⊗ (Σ−1 − S11). (15)Hence,H2 = Tvec(R˜)′[Σ˜−1 ⊗ (Σ˜−1 − S˜11)−1]vec(R˜) d→ χ2m2 . (16)Remark 3. Evidently if, in a given application, the estimated matrix Σ˜−1−S˜11is not at least positive semi-definite, the test fails. If the matrix itself (notonly the estimated one) is positive semi-definite but not positive definite, thedistribution is still asymptotically χ2 , but with degrees of freedom equal to therank Σ−1 − S11 .Remark 4. Note that in the case m = 1 , and consequently R˜ = ρ˜ , the teststatistic of Eq. (16) reduces toTvec(R˜)′[Σ˜−1 ⊗ (Σ˜−1S˜11)−1]vec(R˜) =T ρ˜21− AVar(a˜11), (17)5where Avar( a˜11) is the variance of the limiting distribution of the OLS es-timated coefficient of the lagged dependent variable. Thus, the H2 statisticreduces to the square of the h -statistic, as given by Durbin (1970), becausebasically Σ ⊗ Σ−1 reduces to unity in the case m = 1 . Thus, the case whereΣ−1 − S11 is not at least positive semi-definite corresponds to the casewhere the asymptotic variance in question is equal to or greater than 1. Whenthis is so one should employ an alternative procedure to be derived below.3 An Alternative Test when the H2 Test FailsWhen the H2 statistic yields inadmissible results we may employ the followingprocedure.Write the model in Eq. (1) asY = U−1R+ZC+E =WD+E, E = (t·),W = (U−1, Y−1, X), D = (R′, C ′)′,(18)where we have merely made use of the alternative specification in Eq. (4). Ifwe could observe U−1 we would simply estimate R by OLS and then carry outa test on R as we would with any other OLS-estimated parameter. Since wedo not, we shall use the OLS residuals from the regression of Y on Z . Theestimator thus obtained isD˜ = (W˜ ′W˜ )−1W˜ ′Y = (W˜ ′W˜ )−1W˜ ′WD+(W˜ ′W˜ )−1W˜ ′E, W˜ = (U˜−1, Z). (19)Noting that, under the null, W − W˜ = [Z(C˜ − C), 0] , we find(W˜ ′W˜ )−1W˜ ′W = I + (W˜ ′W˜ )−1W˜ ′[Z(C˜ − C), 0] P→ I.Specifically, note that W − W˜ = (Z(C˜ − C), 0) , so that(W˜ ′W˜ )−1W˜ ′[W − W˜ ]D = 0,because under the null R = 0 . Consequently, under the null,D˜ P→D, and, moreover6√T (D˜ −D) ∼(W˜ ′W˜T)−1 W˜ ′E√T. (20)so that D is estimated consistently and has a well defined limiting distribution.Concentrate now on the estimator of R , viz.√TR˜ = S˜∗11√TT∑t=1w˜′t·t·,(W˜ ′W˜T)−1= S˜∗ =[S˜∗11 S˜∗12S˜∗21 S˜∗22]=[S˜∗1S˜∗2]. (21)Letting S∗ij represent the corresponding blocks in the probability limit of S˜∗ ,and vectorizing the expression in Eq. (21) we find√T vec(R˜) =√T r˜ ∼ (Im ⊗ S∗1)T∑t=2(Im ⊗ w˜′t·)′t·. (22)Since this model too obeys the condition of the Lindeberg theorem, we thereforeconclude √T r˜ d→ N(0,Σ⊗ S∗11). (23)Consequently, we may testH0 : R = 0as against the alternativeH1 : R 6= 0by means of the statisticH∗2 = T r˜′(Σ˜⊗ S˜∗11)−1r˜d→ χ2m2 . (24)4 Diagonal RWhen the autoregression matrix R is diagonal, the situation is more complexthan that of the simple Durbin context, unlessCov(t·) = diag(σ11, σ22, . . . , σmm), (25)7in which case we are reduced to doing m h -tests seriatim.We now examine the case where Σ is unrestricted, i.e. we produce theanalogue of the H2 -statistic when R is diagonal but the elements of ut· arecross correlated. Specifically, the alternative dealt with isut· = ut−1·R + t·, R = diag(r11, r22, . . . , rmm), Cov(t·) = Σ (26)whereΣ = (σij), σij 6= 0, for i 6= j .If the u ’s could be observed, we would write the model asu = V r + e, u = vec(U), V = diag(v·1, v·2, . . . , v·m), , (27)where v·i is the i th column of U−1, r = (r11, r22, . . . , rmm)′ , and estimaterˆ = [V ′(Σ−1 ⊗ IT )V ]−1V ′(Σ−1 ⊗ IT )u; (28)the limiting distribution of the entity above is given by√T rˆ d→ N(0,Ω−1∗ ), Ω∗ = (σijσij). (29)Since they are not known, we may substitute the corresponding OLS residuals,instead of U and U−1 . When we do so we have, under the null,√T r˜ = [V˜ ′(Σ˜−1 ⊗ IT )V˜ ]−1V˜ ′(Σ˜−1 ⊗ IT )u˜ ∼ Ω−1∗1√T[m∑j=1σij v˜′·iu˜·j]. (30)But,1√Tv˜′·iu˜·j ∼1√T(v′·iu·j)−( 1T v′·iY−1)√T (aˆ·j − a·j) ∼1√T(v·i − ZS ′1σ·i)′ u·j.(31)Defining the matrixV ∗ = diag(v·1 − ZS ′1σ·1, v·1 − ZS ′1σ·2, . . . , v·m − ZS ′1σ·m), (32)we may finally write√T r˜ ∼ Ω−1∗1√TV ∗′(Σ−1 ⊗ IT )u d→ N(0,Φ), Φ = Ω−1∗ [σij(σij − σ′·iS11σ·j)]Ω−1∗ .(33)8Remark 5. The matrix Ω∗ is non-singular as the following demonstrationeasily shows. Let e·i be an m -element column vector all of hose elements arezero, except the i th which is one. Then note thatΩ∗ = H ′(Σ−1 ⊗ Σ)H, H = diag(e·1, e·2, . . . , e·m).The non-singularity of Ω∗ follows from the non-singularity of Σ and the factthat H is evidently of rank m . Since the generalized inverse of H and H ′ aregiven respectively byHg = H ′, H ′g = H, because HH ′ = Imit follows thatΩ∗g = Hg(Σ⊗ Σ−1)H ′g = H ′(Σ⊗ Σ−1)H,which is non-singular and, thus, it is the inverse of Ω∗ .If the matrixΩ1 = [σijσij − σijσ′·iS11σ·j]is at least positive semi-definite, we may carry out a test of the null by meansof the test statisticH2D = T r˜′Φ˜−1r˜d→ χ2m, or, more generally, H2Dd→ χ2rank(Ω1). (34)Remark 6. Notice that in the case m = 1 , H2D reduces to the square of theh -statistic because Ω∗ = 1 and Ω1 = 1− Avar(aˆ11) , as in Durbin (1970).If the matrix Ω1 is indefinite, or negative definite, the test above is in-operable and an alternative test may be undertaken as follows. Write (theobservations on) the equations of the model asy·i = riiv·i + Zc·i + ·i, i = 1, 2, 3, . . . , m, (35)and stack them so that the observations on the entire model may be written asy = V r+(Im⊗Z)c+e, r = (r11, r22, r33, . . . , rmm)′, V = diag(v·1, v·2, v·3, . . . , v·m).(36)9Since V is not observable we use instead the columns of the OLS residualsU˜−1 , i.e.V˜ = diag(v˜·1, v˜·2, v˜·3, . . . , v˜·m), (37)to estimated˜ = [(W˜ ′(Σ˜−1⊗IT )W˜ ]−1W˜ ′(Σ˜−1⊗IT )y, W˜ = [V˜ , (Im⊗Z)], d = (r′, c′)′. (38)As in the discussion above, we can show that, under the null,(W − W˜ )d = 0,so the estimator of d , and hence of r , is consistent. Moreover, under the null,√T (d˜− d) ∼ Φ 1√TW ′(Σ−1 ⊗ Im)ed→ N(0,Φ), (39)whereΦ−1 = plimT→∞1TW ′(Σ−1 ⊗ IT )W. (40)It follows then, that under the null,√T r˜ d→ N(0,Φ11), (41)where Φ11 is the m×m principal submatrix of Φ . Consequently, to test thenull that r = 0 we may use the test statisticH∗2D = T r˜′Φ˜−111 r˜d→ χ2m, (42)whereΦ˜−111 =1T(V˜ ′(Σ˜−1 ⊗ IT )V˜ − V˜ ′(Σ˜−1 ⊗ Z(Z ′Z)−1Z ′)V˜), (43)and Σ˜ = U˜ ′U˜/T .If an estimator for r is obtainable, the matrix of Eq. (41) will be positivedefinite and hence it is always operational in practice.10REFERENCESDavidson, R and J. G. MacKinnon (1993), Estimation and Inference in Econo-metrics, New York: Oxford University Press.Dhrymes, P. J. (1989), Topics in Advanced Econometrrics: Probability Founda-tions, New York: Springer-Verlag.Durbin, J. M. (1970), “Testing for Serial Correlation in Least Squares Regressionwhen some of the Regressors are Laggerd Dependent Variables, Econometrica,vol. 38, pp. 410-421.Greene, W. H. (1999), Econometric Analysis, fourth ed., New York: Prentice-Hall.11

Testing for autocorrelation in systems of equations

 Columbia University  Department of Economics Discussion Paper Series           Testing for Autocorrelation in Systems of Equations   Phoebus J. Dhrymes           Discussion Paper No.: 0203-20        Department of Economics Columbia University New York, NY 10027 August 2003  Testing for Autocorrelation in Systems ofEquations∗PHOEBUS J. DHRYMESColumbia UniversityJune 2003AbstractThis paper deals with the problem of testing for the presence of au-tocorrelation in a system of general linear models (Seemingly UnrelatedRegressions, SUR) when the model is formulated as a vector autoregres-sion (VAR) with exogenous variables. The solution presented in thispaper is a generalization of the h-statistic for the single equation singleparameter case given in Durbin (1970).KEY WORDS: Autocorrelation, h-test, H2 -test, VAR1 IntroductionConsider the model∗ c©Phoebus J. Dhrymes, 2003Paper to be presentd at the Magdalinos Memorial Conference, November 20-21, in Athens,Greece.Preliminary material; not to be quoted or disseminated without permission of the author.1yt· = yt−1·A+ xt·B + ut·, t = 1, 2, 3, . . . , T, (1)where yt· is an m -element row vector of dependent, and xt· is a k -element vec-tor of independent variables, respectively; ut·, t = 1, 2, . . . , T is the structuralerror vector. We assumei. {ut· : t = 1, 2, 3, . . . , T} is a sequence of independent identically distrib-uted (i.i.d.) random vectors withE ut· = 0, Cov(ut·) = Σ > 0, (2)defined on some probability space ( Ω , A , P ).ii. It is further assumed thatplimT→∞X ′XT=Mxx > 0, (3)and that the elements in X and U are mutually independent.iii. The system of Eq. (1) is stable, i.e. the characteristic roots of A are lessthan one in absolute value.Regarding the errors, the alternative hypothesis we entertain isut· = ut−1·R + t·. (4)We require, for stationarity, the following assumptions:1. The matrix R is non-singular and stable, i.e. its characteristic roots areless than one in absolute value;2. With little loss of generality, and certainly no loss of relevance, we furtherassume that the matrix R is diagonalizable, i.e. it has the representationR = PΛP−1 , where Λ is the (diagonal) matrix of its characteristic roots.This problem, for the case m = 1 , (and R a scalar) was dealt with by Durbin(1970). A search of widely used econometrics textbooks such as Greene (1999)and Davidson and MacKinnon (1993) discloses no mention of its generalizationto VARs.2Remark 1. If one were to write down a VAR one would normally not beconcerned about the behavior of the “error”, since by definition the errors insuch a system are assumed to be i.i.d. If not, one simply specifies a VAR ofa higher order, in empirical applications. Notwithstanding this observation, inmany applied contexts the logic of the economic model requires the presence ofa specific number of lagged endogenous variables. In such a case, the problemwe are examining here may arise.Remark 2. When the structural error, ut· , is a first order autoregression,the OLS estimators for the parameters of the model in Eq. (1) are inconsis-tent because of the presence of lagged endogenous variables, which aretherefore correlated with the structural error.Thus, if we suspect that the form given in Eq. (4) may be appropriate, we maywish to test the hypothesisH0 : R = 0 ,as against the alternativeH1 : R 6= 0 ,when least squares (OLS) is used to estimate the unknown parameters of Eq.(1).2 Derivation of the Test StatisticWriting the sample asY = Y−1A+XB + U = ZC + U, Z = (Y−1, X), C = (A′, B′)′, (5)the OLS estimator of C is given byC˜ = (Z ′Z)−1Z ′Y = C + (Z ′Z)−1Z ′U. (6)Assuming that a central limit theorem (CLT), such as the Lindeberg CLT, seeDhrymes (1989), pp. 271 ff, we may write the limiting distribution of the OLS3estimator as √T (c˜− c) d→ N(0,Σ⊗M−1zz ), (7)where c = vec(C) and Mzz = plimT→∞[Z ′Z/T ] .From this it is easily verified that√T (a˜− a) d→ N(0,Σ⊗ S11), where M−1zz =[S11 S12S21 S22], (8)and S11 is the (principal) submatrix of M−1zz , consisting of its first m rowsand columns.LetU˜ = Y − ZC˜ = U − Y−1(A˜− A)−X(B˜ −B), (9)be the matrix of OLS residuals and consider the estimator of RR˜ = (U˜ ′−1U˜−1)−1U˜ ′−1U˜ (10)Using Eq. (10), and omitting terms that converge to zero in probability, wemay write, see Dhrymes (1989), pp 161 ff.1√T (R˜− R) ∼ (U˜ ′−1U˜−1)−11√T(U ′−1U − U ′−1Y−1(A˜− A)), (11)either because Z ′Z/T converges, or because C˜ is consistent and has a welldefined limiting distribution, or both. Moreover, using the result again, andbearing in mind that1T U′−1Y−1P→ Σwe finally obtain√T (R˜ −R) ∼ Σ−1 1√TU ′−1U −√T (A˜− A). (12)LetM−1zz =[S11 S12S21 S22]=[S1S2],1The notation X ∼W below means X has the same limiting distribution as W .4and note that √T (A˜− A) ∼ S11√TZ ′U.Vectorizing, we have the expression√Tvec(R˜−R) ∼ 1√T((Im ⊗ [Σ−1U ′−1 − S1Z ′])u, u = vec(U), (13)which obeys the conditions of the Lindeberg CLT, see Dhrymes (1989), pp. 271ff. LetAt = σ(us·, s ≤ t), (14)i.e. it is the σ -algebra generated by the u’s up to t . To evaluate the covariancematrix of the limiting distribution of the left member of Eq. (13), we need tofind the expectation of terms like (Im⊗K)uu′(Im⊗K) . We shall do so by firstconditioning with respect to At−1 . Thus we need to evaluateEAt−1E([Im ⊗ (Σ−1U ′−1 − S1Z ′)]uu′[Im ⊗ (U−1Σ−1 − ZS ′1)]|At−1)=EAt−1(Σ⊗ [Σ−1U ′−1U−1Σ−1 − Σ−1U ′−1ZS ′1 − S1Z ′U−1Σ−1 + S1Z ′ZS ′1])=Σ⊗ [Σ−1ΣΣ−1 − Σ−1(Σ, 0)S ′1 − S1(Σ, 0)′Σ−1 + S11]=Σ⊗ (Σ−1 − S11). (15)Hence,H2 = Tvec(R˜)′[Σ˜−1 ⊗ (Σ˜−1 − S˜11)−1]vec(R˜) d→ χ2m2 . (16)Remark 3. Evidently if, in a given application, the estimated matrix Σ˜−1−S˜11is not at least positive semi-definite, the test fails. If the matrix itself (notonly the estimated one) is positive semi-definite but not positive definite, thedistribution is still asymptotically χ2 , but with degrees of freedom equal to therank Σ−1 − S11 .Remark 4. Note that in the case m = 1 , and consequently R˜ = ρ˜ , the teststatistic of Eq. (16) reduces toTvec(R˜)′[Σ˜−1 ⊗ (Σ˜−1S˜11)−1]vec(R˜) =T ρ˜21− AVar(a˜11), (17)5where Avar( a˜11) is the variance of the limiting distribution of the OLS es-timated coefficient of the lagged dependent variable. Thus, the H2 statisticreduces to the square of the h -statistic, as given by Durbin (1970), becausebasically Σ ⊗ Σ−1 reduces to unity in the case m = 1 . Thus, the case whereΣ−1 − S11 is not at least positive semi-definite corresponds to the casewhere the asymptotic variance in question is equal to or greater than 1. Whenthis is so one should employ an alternative procedure to be derived below.3 An Alternative Test when the H2 Test FailsWhen the H2 statistic yields inadmissible results we may employ the followingprocedure.Write the model in Eq. (1) asY = U−1R+ZC+E =WD+E, E = (t·),W = (U−1, Y−1, X), D = (R′, C ′)′,(18)where we have merely made use of the alternative specification in Eq. (4). Ifwe could observe U−1 we would simply estimate R by OLS and then carry outa test on R as we would with any other OLS-estimated parameter. Since wedo not, we shall use the OLS residuals from the regression of Y on Z . Theestimator thus obtained isD˜ = (W˜ ′W˜ )−1W˜ ′Y = (W˜ ′W˜ )−1W˜ ′WD+(W˜ ′W˜ )−1W˜ ′E, W˜ = (U˜−1, Z). (19)Noting that, under the null, W − W˜ = [Z(C˜ − C), 0] , we find(W˜ ′W˜ )−1W˜ ′W = I + (W˜ ′W˜ )−1W˜ ′[Z(C˜ − C), 0] P→ I.Specifically, note that W − W˜ = (Z(C˜ − C), 0) , so that(W˜ ′W˜ )−1W˜ ′[W − W˜ ]D = 0,because under the null R = 0 . Consequently, under the null,D˜ P→D, and, moreover6√T (D˜ −D) ∼(W˜ ′W˜T)−1 W˜ ′E√T. (20)so that D is estimated consistently and has a well defined limiting distribution.Concentrate now on the estimator of R , viz.√TR˜ = S˜∗11√TT∑t=1w˜′t·t·,(W˜ ′W˜T)−1= S˜∗ =[S˜∗11 S˜∗12S˜∗21 S˜∗22]=[S˜∗1S˜∗2]. (21)Letting S∗ij represent the corresponding blocks in the probability limit of S˜∗ ,and vectorizing the expression in Eq. (21) we find√T vec(R˜) =√T r˜ ∼ (Im ⊗ S∗1)T∑t=2(Im ⊗ w˜′t·)′t·. (22)Since this model too obeys the condition of the Lindeberg theorem, we thereforeconclude √T r˜ d→ N(0,Σ⊗ S∗11). (23)Consequently, we may testH0 : R = 0as against the alternativeH1 : R 6= 0by means of the statisticH∗2 = T r˜′(Σ˜⊗ S˜∗11)−1r˜d→ χ2m2 . (24)4 Diagonal RWhen the autoregression matrix R is diagonal, the situation is more complexthan that of the simple Durbin context, unlessCov(t·) = diag(σ11, σ22, . . . , σmm), (25)7in which case we are reduced to doing m h -tests seriatim.We now examine the case where Σ is unrestricted, i.e. we produce theanalogue of the H2 -statistic when R is diagonal but the elements of ut· arecross correlated. Specifically, the alternative dealt with isut· = ut−1·R + t·, R = diag(r11, r22, . . . , rmm), Cov(t·) = Σ (26)whereΣ = (σij), σij 6= 0, for i 6= j .If the u ’s could be observed, we would write the model asu = V r + e, u = vec(U), V = diag(v·1, v·2, . . . , v·m), , (27)where v·i is the i th column of U−1, r = (r11, r22, . . . , rmm)′ , and estimaterˆ = [V ′(Σ−1 ⊗ IT )V ]−1V ′(Σ−1 ⊗ IT )u; (28)the limiting distribution of the entity above is given by√T rˆ d→ N(0,Ω−1∗ ), Ω∗ = (σijσij). (29)Since they are not known, we may substitute the corresponding OLS residuals,instead of U and U−1 . When we do so we have, under the null,√T r˜ = [V˜ ′(Σ˜−1 ⊗ IT )V˜ ]−1V˜ ′(Σ˜−1 ⊗ IT )u˜ ∼ Ω−1∗1√T[m∑j=1σij v˜′·iu˜·j]. (30)But,1√Tv˜′·iu˜·j ∼1√T(v′·iu·j)−( 1T v′·iY−1)√T (aˆ·j − a·j) ∼1√T(v·i − ZS ′1σ·i)′ u·j.(31)Defining the matrixV ∗ = diag(v·1 − ZS ′1σ·1, v·1 − ZS ′1σ·2, . . . , v·m − ZS ′1σ·m), (32)we may finally write√T r˜ ∼ Ω−1∗1√TV ∗′(Σ−1 ⊗ IT )u d→ N(0,Φ), Φ = Ω−1∗ [σij(σij − σ′·iS11σ·j)]Ω−1∗ .(33)8Remark 5. The matrix Ω∗ is non-singular as the following demonstrationeasily shows. Let e·i be an m -element column vector all of hose elements arezero, except the i th which is one. Then note thatΩ∗ = H ′(Σ−1 ⊗ Σ)H, H = diag(e·1, e·2, . . . , e·m).The non-singularity of Ω∗ follows from the non-singularity of Σ and the factthat H is evidently of rank m . Since the generalized inverse of H and H ′ aregiven respectively byHg = H ′, H ′g = H, because HH ′ = Imit follows thatΩ∗g = Hg(Σ⊗ Σ−1)H ′g = H ′(Σ⊗ Σ−1)H,which is non-singular and, thus, it is the inverse of Ω∗ .If the matrixΩ1 = [σijσij − σijσ′·iS11σ·j]is at least positive semi-definite, we may carry out a test of the null by meansof the test statisticH2D = T r˜′Φ˜−1r˜d→ χ2m, or, more generally, H2Dd→ χ2rank(Ω1). (34)Remark 6. Notice that in the case m = 1 , H2D reduces to the square of theh -statistic because Ω∗ = 1 and Ω1 = 1− Avar(aˆ11) , as in Durbin (1970).If the matrix Ω1 is indefinite, or negative definite, the test above is in-operable and an alternative test may be undertaken as follows. Write (theobservations on) the equations of the model asy·i = riiv·i + Zc·i + ·i, i = 1, 2, 3, . . . , m, (35)and stack them so that the observations on the entire model may be written asy = V r+(Im⊗Z)c+e, r = (r11, r22, r33, . . . , rmm)′, V = diag(v·1, v·2, v·3, . . . , v·m).(36)9Since V is not observable we use instead the columns of the OLS residualsU˜−1 , i.e.V˜ = diag(v˜·1, v˜·2, v˜·3, . . . , v˜·m), (37)to estimated˜ = [(W˜ ′(Σ˜−1⊗IT )W˜ ]−1W˜ ′(Σ˜−1⊗IT )y, W˜ = [V˜ , (Im⊗Z)], d = (r′, c′)′. (38)As in the discussion above, we can show that, under the null,(W − W˜ )d = 0,so the estimator of d , and hence of r , is consistent. Moreover, under the null,√T (d˜− d) ∼ Φ 1√TW ′(Σ−1 ⊗ Im)ed→ N(0,Φ), (39)whereΦ−1 = plimT→∞1TW ′(Σ−1 ⊗ IT )W. (40)It follows then, that under the null,√T r˜ d→ N(0,Φ11), (41)where Φ11 is the m×m principal submatrix of Φ . Consequently, to test thenull that r = 0 we may use the test statisticH∗2D = T r˜′Φ˜−111 r˜d→ χ2m, (42)whereΦ˜−111 =1T(V˜ ′(Σ˜−1 ⊗ IT )V˜ − V˜ ′(Σ˜−1 ⊗ Z(Z ′Z)−1Z ′)V˜), (43)and Σ˜ = U˜ ′U˜/T .If an estimator for r is obtainable, the matrix of Eq. (41) will be positivedefinite and hence it is always operational in practice.10REFERENCESDavidson, R and J. G. MacKinnon (1993), Estimation and Inference in Econo-metrics, New York: Oxford University Press.Dhrymes, P. J. (1989), Topics in Advanced Econometrrics: Probability Founda-tions, New York: Springer-Verlag.Durbin, J. M. (1970), “Testing for Serial Correlation in Least Squares Regressionwhen some of the Regressors are Laggerd Dependent Variables, Econometrica,vol. 38, pp. 410-421.Greene, W. H. (1999), Econometric Analysis, fourth ed., New York: Prentice-Hall.11

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Testing for autocorrelation in systems of equations

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